Add Folders to Path

We start by adding the necessary folders to the current working path.

Overview

In the following notebok we compute the TRA from AVISO data. The notebook is structured as follows:

  1. Import data from the file 'AVISO.mat' stored in the folder 'data'
  2. Define computational parameters (such as the number of cores) and data
  3. Define spatio-temporal domain.
  4. Interpolate velocity from (discrete) gridded data
  5. Trajectory Rotation Angle TRA¯:

    • Compute velocity along trajectories {x˙(ti)}i=0N over the time interval [t0,tN].

    • Compute TRA¯ from velocity along trajectories:

      (1)TRA¯t0tN=1tNt0i=0N1cos1x˙(ti),x˙(ti+1)+v02|x˙(ti)|2+v02|x˙(ti+1)|2+v02

      v0 is a characteristic velocity, which can be be estimated by taking the spatio-temporal average of the velocity over the whole dataset.

    • Finally we test the ability of the TRA¯ to detect vortical flow features in a sparse data setting by progressively downsampling the data.

Import Data

Computational parameters and data

Here we define the computational parameters and the data.

Spatio-temporal domain

Here we define the spatio-temporal domain over which to consider the dynamical system.

Velocity interpolation

In order to evaluate the velocity field at arbitrary locations and times, we must interpolate the discrete velocity data. The interpolation with respect to time is always linear. The interpolation with respect to space can be chosen to be "cubic" or "linear". In order to favour a smooth velocity field, we interpolate the velocity field in space using a cubic interpolant.

Trajectory Rotation Average (TRA¯)

Regions of high TRA¯t0tN are linked to vortical flow structures. Note that the TRA¯ highlights the same vortical flow areas as other commonly used rotation diagnostics such as the Polar Rotation Angle or the EllipticLCS. In order to test the ability of the TRA¯ to extract vortical flow regions even in a sparse data setting, we randomly progressively downsample data. As shown in the following figures, TRA¯ is able to capture the three major lagrangian eddies even at very low resolutions.

References

[1] Haller, G., Aksamit, N., & Encinas-Bartos, A. P. (2021). Quasi-objective coherent structure diagnostics from single trajectories. Chaos: An Interdisciplinary Journal of Nonlinear Science, 31(4), 043131.